Question
TU
is tangent to ⨀S. What is m∠U?
S
T
U
56°
m∠U=
°
is tangent to ⨀S. What is m∠U?
S
T
U
56°
m∠U=
°
Answers
GPT-4o mini
To solve for \( m\angle U \) when line \( TU \) is tangent to circle \( S \) and forms a tangent-chord angle with chord \( TS \) that subtends arc \( TS \), we can use the tangent-chord angle theorem.
The theorem states that the measure of the angle formed by a tangent and a chord through the point of contact is equal to half the measure of the intercepted arc.
Given:
- \( m\angle TUS = 56^\circ \)
The relationship can be expressed as:
\[
m\angle U = \frac{1}{2} \times m\text{(arc TS)}
\]
We need to determine the arc length \( TS \):
Since the angle \( \angle U \) is on the exterior of the circle, we can say:
\[
m\text{(arc TS)} = 2 \times m\angle U
\]
However, since we only have the angle \( \angle TUS \) given (which helps identify the intercepting arc), it applies directly as the tangent angle causing the intercept of other angles involved.
Since \( m\angle TUS \) and arc \( TS \) are related in a circle, if \( m\angle TUS = 56^\circ \), \( m\angle U \) equals the tangent angle as measured directly resulting in:
\[
m\angle U = 56^\circ
\]
Thus, the measure \( m\angle U \) is:
\[
\boxed{56}
\]
In conclusion, if \( \angle U \) indeed overlaps \( \angle TUS \), and directly subtends with no intermediary angle, thus resulting as such. However, if arcs are different or adjustments needed based on physical diagrams, let's adjust accordingly.
The theorem states that the measure of the angle formed by a tangent and a chord through the point of contact is equal to half the measure of the intercepted arc.
Given:
- \( m\angle TUS = 56^\circ \)
The relationship can be expressed as:
\[
m\angle U = \frac{1}{2} \times m\text{(arc TS)}
\]
We need to determine the arc length \( TS \):
Since the angle \( \angle U \) is on the exterior of the circle, we can say:
\[
m\text{(arc TS)} = 2 \times m\angle U
\]
However, since we only have the angle \( \angle TUS \) given (which helps identify the intercepting arc), it applies directly as the tangent angle causing the intercept of other angles involved.
Since \( m\angle TUS \) and arc \( TS \) are related in a circle, if \( m\angle TUS = 56^\circ \), \( m\angle U \) equals the tangent angle as measured directly resulting in:
\[
m\angle U = 56^\circ
\]
Thus, the measure \( m\angle U \) is:
\[
\boxed{56}
\]
In conclusion, if \( \angle U \) indeed overlaps \( \angle TUS \), and directly subtends with no intermediary angle, thus resulting as such. However, if arcs are different or adjustments needed based on physical diagrams, let's adjust accordingly.